Here is the happy shortcut that the limit laws hand us. For most of the friendly functions you meet — polynomials, and ratios where the bottom isn't zero — you can find a limit by simply plugging in:

No table, no graph. Just substitute x = c and evaluate. Functions where this works are called continuous at c — the curve has no break there, so where it heads is exactly where it is.

Why it works

Take a polynomial like f(x) = x^2 - 3x + 5. Build it from the limit laws: \lim x = c and \lim k = k, then the power, product, and sum laws stitch the pieces together. The result is just the polynomial with c substituted in:

Slide x toward c and watch the point ride the smooth curve straight to f(c) — no surprises.

Plug in and read off

The curve below is f(x) = x^2 - 3x + 5. The dashed guides meet exactly at (2, 3) — substitution and the graph agree.

When you may NOT just plug in

Substitution is only valid where the function is continuous. If plugging in gives a nonzero number over zero — like \lim_{x \to 0} \tfrac{1}{x} — the function blows up and the limit fails. And if it gives the form \tfrac{0}{0}, the answer is hiding: that's an indeterminate form, and we'll need algebra (factoring) to dig it out. Substitution is the first thing to try — just check what you get.

Watch Sal substitute